Let's solve these questions one by one. These are all about finding the equation of a line that passes through two given points. The general form of the equation of a line is:

$y = mx + b$

Where:

• $m$ is the slope of the line.
• $b$ is the y-intercept.

To find $m$ (slope), we use the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Once the slope is found, we can substitute one point into the equation to solve for $b$.

1) What is an equation for the line that passes through the coordinates (2,7) and (0,1)?

Step 1: Find the slope

Using the points (2,7) and (0,1):

$m = \frac{7 - 1}{2 - 0} = \frac{6}{2} = 3$

Step 2: Find the equation

Substitute the point (0,1) into $y = 3x + b$:

$1 = 3(0) + b \quad \Rightarrow \quad b = 1$

So, the equation of the line is:

$y = 3x + 1$

2) What is an equation for the line that passes through the coordinates (2,0) and (0,3)?

Step 1: Find the slope

Using the points (2,0) and (0,3):

$m = \frac{0 - 3}{2 - 0} = \frac{-3}{2}$

Step 2: Find the equation

Substitute the point (0,3) into $y = -\frac{3}{2}x + b$:

$3 = -\frac{3}{2}(0) + b \quad \Rightarrow \quad b = 3$

So, the equation of the line is:

$y = -\frac{3}{2}x + 3$

3) What is an equation for the line that passes through the coordinates (-1,2) and (7,6)?

Step 1: Find the slope

Using the points (-1,2) and (7,6):

$m = \frac{6 - 2}{7 - (-1)} = \frac{4}{8} = \frac{1}{2}$

Step 2: Find the equation

Substitute the point (-1,2) into $y = \frac{1}{2}x + b$:

$2 = \frac{1}{2}(-1) + b \quad \Rightarrow \quad 2 = -\frac{1}{2} + b \quad \Rightarrow \quad b = 2 + \frac{1}{2} = \frac{5}{2}$

So, the equation of the line is:

$y = \frac{1}{2}x + \frac{5}{2}$

4) Find the equation of the line that passes through the points (1,1) and (3,5).

Step 1: Find the slope

Using the points (1,1) and (3,5):

$m = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2$

Step 2: Find the equation

Substitute the point (1,1) into $y = 2x + b$:

$1 = 2(1) + b \quad \Rightarrow \quad b = 1 - 2 = -1$

So, the equation of the line is:

$y = 2x - 1$

5) Find the equation of the line that passes through the points (1,3) and (2,4).

Step 1: Find the slope

Using the points (1,3) and (2,4):

$m = \frac{4 - 3}{2 - 1} = \frac{1}{1} = 1$

Step 2: Find the equation

Substitute the point (1,3) into $y = x + b$:

$3 = 1(1) + b \quad \Rightarrow \quad b = 3 - 1 = 2$

So, the equation of the line is:

$y = x + 2$

Would you like any further details on these steps?

Here are 5 related questions to explore further:

1. How do you derive the slope formula for a line between two points?
2. What is the geometric interpretation of a slope of 0 or an undefined slope?
3. How do you find the equation of a line parallel to a given line?
4. How do you find the equation of a line perpendicular to a given line?
5. How can you convert a linear equation into slope-intercept form from point-slope form?

Tip: Remember, the slope $m$ tells you how steep the line is. A positive slope means the line rises as you move right, while a negative slope means it falls.